WASHINGTON—Former FBI Director James Comey revealed in a series of private memos that President Donald Trump and his then-chief of staff had doubts within days of taking office about national security adviser Mike Flynn, who subsequently left the administration after misleading officials about his contacts with Russia and later pleaded guilty to lying to law enforcement.

Mr. Comey’s previously unreported account of their take on Mr. Flynn was part of seven memos spanning 15 pages that were authored by Mr. Comey over a four-month...